2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved S0294-1449(01)00071-3/FLA
ON SOME PERIODIC HARTREE-TYPE MODELS FOR CRYSTALS
I. CATTOa, C. LE BRISb, P.-L. LIONSa
aCEREMADE, CNRS UMR 7534, Université Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris cedex 16, France
bCERMICS, Ecole Nationale des Ponts et Chaussées, 6 & 8, avenue Blaise Pascal, Cité Descartes, Champs sur Marne, 77455 Marne-La-Vallée cedex, France
Received 28 September 2000
ABSTRACT. – We continue here our study of the thermodynamic limit for various models of Quantum Chemistry. More specifically, we study the Hartree and the restricted Hartree model.
For the restricted Hartree model, we prove the existence of the thermodynamic limit for the energy per unit volume. We also define a periodic problem associated to the Hartree model, and we prove that it is well-posed.2002 Éditions scientifiques et médicales Elsevier SAS
RÉSUMÉ. – Nous poursuivons dans cet article notre étude systématique de la limite thermodynamique de divers modèles issus de la Chimie Quantique Moléculaire. Nous étudions plus spécifiquement les modèles de Hartree et de Hartree restreint. Pour le modèle de Hartree restreint, nous prouvons l’existence de la limite thermodynamique de l’énergie par unité de volume. Nous définissons également un modèle périodique associé au modèle de Hartree, et nous démontrons qu’il est bien posé.2002 Éditions scientifiques et médicales Elsevier SAS
1. Introduction
We consider here the thermodynamic limit (or bulk limit) problem for some Hartree type models, thereby continuing a long term work that we have begun in [11] with a similar study in the setting of the Thomas–Fermi–von Weizsäcker type models. The results we have obtained in that framework were summarized in [10], those we shall obtain here have been announced in [12]. It is to be mentioned that we also consider in [13] the same problem for the reduced Hartree–Fock and the Hartree–Fock models.
For the sake of consistency, we briefly recall now the motivations of our work. We also say a few words on how this work interacts with other mathematical studies. And we refer the reader to [11] for a more detailed introduction.
The present work, as well as our previous ones, finds its roots in many mathematical studies devoted to the mathematical counterpart of problems of Statistical Mechanics.
E-mail address: catto@ceremade.dauphine.fr (I. Catto).
Briefly speaking, the so-called thermodynamic limit problem consists of examining the behaviour of models for a finite volume of matter when the volume under consideration goes to infinity. Since the energy is an extensive thermodynamic quantity, it is expected that the energy per unit volume goes to a finite limit when the volume goes to infinity.
It is also expected that the function representing the state of the matter goes also to a limit in some sense. To fix the ideas, let us make precise these questions in the case of an infinite crystal and in the setting of a model of the density functional theory. We shall see extensions of this simplified setting later on.
Consider a finite number of nuclei, each nucleus being of unit charge and being located at a pointk=(k1, k2, k3)of integral coordinates in R3, which is the center of a cubic unit cellQk= {(x1, x2, x3)∈R3; −12< xi−ki 12, i=1,2,3}(with the convention thatQ0
will be henceforth denoted byQ). The set of the positions of these nuclei is then a finite subset of the set of all points of integral coordinates that is Z3⊂R3. The union of all cubic cells whose center is a point of is denoted by( ); its volume is denoted by
| |. Since each cell has unit volume and each nucleus is of unit charge,| |is also the number of nuclei and the total nuclear charge.
Suppose that for ⊂Z3 fixed, we have a well-posed model for the ground state of the neutral molecule consisting of | | electrons and | | nuclei located at the points of . Let us denote byI the ground-state energy, and byρ the minimizing electronic density.
Then, the question of the existence of the thermodynamic (or bulk) limit for the model under consideration may be stated as follows:
(i) Does there exist a limit for the energy per unit volume |1|I when| |goes to infinity?
(ii) Does the minimizing density ρ approach a limit ρ∞ (in a sense to be made precise later) when| |goes to infinity?
(iii) Does the limit densityρ∞have the same periodicity as the assumed periodicity of the nuclei?
We shall not deal here with the physical background of this theoretical problem, and we refer the reader to the textbooks [4,51] and articles [26,30]. We prefer to concentrate ourselves on the mathematical works that are devoted to this difficult question.
The models we shall consider are models arising in Quantum Chemistry, and therefore models that are only valid at zero temperature. From the mathematical viewpoint, the thermodynamic limit problem has been extensively studied, in the zero temperature case as well as in the case of strictly positive temperatures.
A brief historical survey should go as follows. The story has really begun with Fischer and Ruelle, who have proved the existence of the thermodynamic limit for the (classical or quantum) microcanonical, canonical, and grand canonical ensembles for a system of particles in Rd (see [44], and references therein). It is worth noticing that their proof did not cover the case of a long range interaction like the Coulomb interaction. It is only in the late sixties that Lieb and Lebowitz, using a result by Dyson and Lenard, proved the existence of the thermodynamic limit for real matter, i.e. with Coulomb forces (see [25–27,24]). This undoubtedly constitutes the first milestone of the mathematical understanding of these problems of Statistical Mechanics. The proof has next been extended by Lieb and Narnhofer [31] in 1974 to deal with the case of Jellium, that is
to say to deal with a model where the electrons are immersed in a uniformly positively charged background.
In 1985, Fefferman laid the second milestone by proving in [17] the existence of the thermodynamic limit for a crystal, in the statistical setting. For the first time, a statistical model of a non spherically symmetric matter was treated in this respect. With slight modifications, Fefferman’s proof has been extended by Gregg in 1989 ([18]) to treat Coulomb-like interactions. Let us emphasize that the two main difficulties that we have just identified, namely the long range nature of the Coulomb potential and the (obvious) lack of spherical symmetry of the periodic lattices, will be of course also present in our work.
In this very brief survey, we have on purpose omitted to mention the ground-breaking work [32] by Lieb and Simon on the thermodynamic limit in the framework of the Thomas–Fermi theory (TF Theory for short). Indeed, this work is at the origin of our own study [11] on the Thomas–Fermi–von Weizsäcker model (TFW model for short), and has therefore a far larger impact on our work than the, however fundamental, works that we have quoted above.
At this stage of our short presentation of the state of the art of the mathematical knowledge on thermodynamic limit problems, we find it useful to briefly recall now the results that we have obtained in [11] on the Thomas–Fermi–von Weizsäcker model.
Indeed, many of the concepts and techniques that we have used in [11] (some of them being inherited from Lieb and Simon, some others being especially introduced by us in order to treat the TFW case) will be useful here. Moreover, recalling the complete results we have obtained in the TFW case will help the reader to place the results we shall obtain here on the Hartree model in this context. It is also to be remarked that our results on the TFW model include Lieb and Simon results on the TF model (suppress simply the gradient term in the energy functional and make the quite obvious corresponding modifications in the sequel).
The Thomas–Fermi–von Weizsäcker model for the neutral molecular system de- scribed above is an improved form of the standard Thomas–Fermi model, and reads as follows
ITFW=inf
ETFW(ρ)+1 2
y=z∈
1
|y−z|;ρ0, √
ρ∈H1R3,
R3
ρ= | |
, (1)
ETFW(ρ)=
R3
∇√ ρ2+
R3
ρ5/3−
R3 k∈
1
|x−k| ρ(x)dx +1
2
R3×R3
ρ(x)ρ(y)
|x−y| dxdy. (2)
The TFW (and as well as the TF) model belongs to a large class of models that is today identified as the models arising in Density Functional Theory: we refer the reader to [14,41] for an introduction to the general features and the physical foundations of such models. Mathematically, it is a well-known fact that the problem (1)–(2) has a unique
minimizing density, denoted by ρ (see Lieb [29], Benguria et al. [5], or Lions [35]), and that, denotingu = √ρ ,u is a solution to
−u + 5
3ρ2/3−
u = −θ u , (3)
where we denote by
=
k∈
1
|x−k|−ρ 1
|x|,
the effective potential the electrons experience, and where θ >0 is the Lagrange multiplier associated to the constraint in (1).
In our previous work [11], we have proved that the three questions (i)–(ii)–(iii) of the thermodynamic limit problem that we have asked above can be answered positively in the setting of the TFW theory. More precisely, let us first of all introduce the periodic potentialGuniquely defined by
−G=4π
−1+
y∈Z3
δ(· −y) , (4)
and
Q
G=0, (5)
and then define the following periodic minimization problem set on the unit cellQ of the lattice
IperTFW=inf
EperTFW(ρ); ρ0, √
ρ∈Hper1 (Q),
Q
ρ=1
, (6)
EperTFW(ρ)=
Q
∇√ ρ2+
Q
ρ5/3−
Q
ρ(x)G(x)dx +1
2Q×Q
ρ(x)ρ(y)G(x−y)dxdy, (7) where
Hper1 (Q)=u∈Hloc1 R3, uperiodic inxi, i=1,2,3, of period 1.
The main results we obtain in [11] may be stated as follows (we need technical assumptions that are irrelevant in this introduction and that we therefore do not make precise here): up to an additive constantM/2 that only depends onGthrough
M=lim
x→0G(x)− 1
|x|, (8)
and which is just a matter of normalization, we have convergence of the TFW energy per unit volume to the infimumIperTFW; moreover the density ρ minimizingITFW converges (uniformly locally, at least) to the unique periodic densityρperminimizingIperTFW.
In view of these results, the reader may understand the main two motivations of our whole work. Our purpose is twofold: first, we want to check that the molecular model under consideration does have the good behaviour in the limit of large volumes;
second, we wish to set a limit problem that is well-posed mathematically and that can be justified in the most possible rigorous way (in particular with a view to give a sound ground to the numerical simulations of the condensed phase). As far as this second aim is concerned, it is clear (at least we hope it is) from the above formulae that one keypoint for the definition of the periodic problem is the definition of laws of interaction between particles, i.e. of the interaction potential(s). In the TFW setting, the second aim was less prominent since the potential G is the same as the one appearing in the TF setting and the periodic minimization problem is rather easy to guess in view of the one arising for the TF theory. Likewise, it is easy to check that this periodic minimization problem is mathematically well-posed. In other words, taking benefit from the work by Lieb and Simon who had already defined the TF periodic problem, the idea to introduce the periodic problem (6)–(7) was straightforward. In [11], our “only” contribution was therefore to prove that the TFW model does converge in the thermodynamic limit to (6)–
(7). The purpose of the present work is the study of the thermodynamic limit problem in the Hartree setting. We shall see below that the guess on the periodic problem is not so obvious in the Hartree model. Consequently, the mere definition of the limit problem turns out to be a substantial piece of the work (writing a periodic problem that has some rigorous mathematical sense is not straightforward). This paper is aimed at describing it.
It will certainly be rather clear to the reader that the questions we tackle here in trying to define as rigorously as possible periodic problems in the Hartree framework are indeed close to questions of interest in Solid State Physics, both for theoretical and numerical purposes. For the sake of brevity, we shall not detail here the relationship between our work and Solid State Physics. We only mention some references here, namely [23,40], and also [2,4,9,39,42,47,48,53], and refer the reader to some future work of our own.
Because of the complexity of the Hartree setting, we shall not be able to do in this setting everything we did in the TFW setting, namely proving the convergence of the energy per unit volume in the thermodynamic limit. We shall indeed prove the convergence of the energy per unit volume in the thermodynamic limit for a simplified Hartree model (namely the restricted Hartree model, treated in Section 3). Furthermore, we shall prove the convergence of the energy per unit volume for one very peculiar form of the true Hartree model (see Section 4), but our efforts to prove it for the generic form of the Hartree model have failed so far. From the single example we have in hand, and from more general considerations, we shall however deduce a general form for a periodic Hartree problem that is likely to be the thermodynamic limit of the Hartree model. We shall prove, still in Section 4, that this periodic model defines a mathematically well- posed minimization problem.
Let us finally mention that the Hartree–Fock setting is discussed by the authors in [13].
But before all, let us devote Section 2 to the definition of the general setting we shall work in, and to the detailed presentation of the results we shall establish.
2. General setting of the models and main results
Let us begin this section by defining the molecular models we shall deal with in this article. There are two of them, namely on the one hand the Hartree model and, on the other hand, its simplified form, the restricted Hartree model. For the sake of brevity, we shall often abbreviate these models in the H and the RH models, respectively.
We recall from the introduction that, for each , finite subset of Z3⊂R3, we consider the molecular system consisting of the | |nuclei located at the points of , and | | electrons. We shall henceforth denote by
V (x)=
k∈
1
|x−k|, (9)
the attraction potential created by the nuclei on the electrons, and by 1
2U =1 2
m,n∈ , m=n
1
|m−n| (10)
the self-repulsion of the nuclei.
As in [11], we shall also consider the case when the nuclei are not point nuclei but are smeared nuclei. In that case, each Dirac mass located at a pointkof is replaced by a compactly supported smooth non-negative function of total mass one, typically denoted bym(· −k), and “centered” at that point of . The regularity of the functionmdoes not play a great role in the sequel, and therefore we shall assume without loss of generality thatmisC∞. The potential (9) and the repulsion (10) are then respectively replaced by
Vm(x)=
k∈
m 1
|x−k|, (11)
1
2Um=1 2D
k∈
m(· +k),
k∈
m(· +k) −1
2| |D(m, m). (12) In the above equation, we have as usual denoted byD(·,·)the double integral defined as follows
D(f, f )=
R3
R3
f (x)f (y)
|x−y| dxdy. (13) It will be convenient to introduce in this setting the function
m =
k∈
m(· −k). (14)
In this setting of smeared nuclei, we shall also make use of the effective potential defined for each electronic densityρ as follows
=(m −ρ ) 1
|x|. (15)
It is now time to recall the properties of the sequence of sets that we shall consider.
For the sake of completeness, we recall here the following definition taken from [11].
DEFINITION 1. – We shall say that a sequence( i)i1of finite subsets of Z3goes to infinity if the following two conditions hold:
(a) For any finite subsetA⊂Z3, there existsi∈N such that
∀ji, A⊂ j.
(b) If h is the set of points in R3whose distance to∂( )is less than h, then
ilim→∞
| hi|
| i| =0, ∀h >0.
Condition (b) will be hereafter referred to as the Van Hove condition.
Briefly speaking, a sequence satisfying the Van Hove condition is a sequence for which the ‘boundary’ is negligible in front of the ‘interior’. A sequence of large cubes typically satisfies the conditions of Definition 1. We shall only consider henceforth Van Hove sequences going to infinity in the sense of the above definition. Following the notation of [32,11], we shall write henceforth lim →∞f ( )instead of limi→∞f ( i).
We now need to define the following useful functional transformation, that we have already used in [11], and which will be again very efficient in the present work.
DEFINITION 2. – For a given sequence and a sequenceρ of densities, we call the∼transform ofρ and denote byρ˜ the following sequence of functions
˜ ρ = 1
| |
k∈
ρ (· +k).
We finally introduce
f (x)= 1
|x|−
Q
dy
|x−y|, next
f (x)=
k∈
1
|x−k|−
Q
dy
|x−k−y| . (16) It is convenient to rewritef as
f =V −χ( ) 1
|x|, (17)
where, more generally, we shall denote byχ*the characteristic function of the domain
*. Besides, it is proved in [32], and recalled in [11], that, whenQis a cube,
|f (x)| C
|x|4 (18)
almost everywhere on R3, for some positive constant C, and that f converges to the periodic potential G+d, for some real constant d independent of , uni- formly on compact subsets of R3\Z3. Moreover, for any compact subset K of R3, f −k∈ ∩K 1
|x−k| converges uniformly onKtoG+d−k∈Z3∩K 1
|x−k| (see [32]).
We shall make use in the sequel of the following notation. IfH is a functional space, we denote byHunif(R3)the space
Hunif
R3=ψ∈DR3/ψ∈H (x+Q)∀x∈R3,sup
x∈R3
ψH (x+Q)<∞.
In addition, we shall also simply writef Qginstead off (χQg).
We are now in position to introduce the molecular models we shall deal with. The Hartree model is defined as follows.
IH =inf
EH(ϕ1;. . .;ϕ| |)+1
2U ;ϕi ∈H1R3,
R3
ϕi2=1, 1i| |
, (19)
EH(ϕ1;. . .;ϕ| |)=| |
i=1 R3
|∇ϕi|2−1
2D|ϕi|2,|ϕi|2 −
R3
V ρ+1
2D(ρ , ρ), (20) with
ρ=| |
i=1
|ϕi|2. (21)
The Hartree model was historically introduced by Hartree in [19]. It is a well-known fact that, for any subset of R3, this minimization problem is attained by at least one vector (ϕ1;. . .;ϕ| |), with ϕi >0 for every 1i| | (see the works by Lieb and Simon in [33] and by Lions in [35]).
In the smeared nuclei case, the energy functional of the Hartree model reads as follows Em,H(ϕ1;. . .;ϕ| |)=| |
i=1 R3
|∇ϕi|2−1
2D|ϕi|2,|ϕi|2 −
R3
Vmρ+1
2D(ρ , ρ), (22) and the minimization problem can therefore be written in the following more concise form
Im,H=inf | |
i=1 R3
|∇ϕi|2−1
2D|ϕi|2,|ϕi|2 +1
2D(ρ−m , ρ−m )
−1
2| |D(m, m);ϕi ∈H1R3,
R3
ϕi2=1,1i| |
, (23) where we recall thatm is given by (14).
As announced above, we also define the restricted Hartree model, obtained from the standard Hartree model by introducing the self-interaction between electroniand itself
in the energy functional. In the point nuclei case, this model reads IRH=inf
ERH(ϕ1;. . .;ϕ| |)+1
2U ;ϕi ∈H1R3,
R3
ϕi2 =1,1i| |
, (24)
ERH(ϕ1;. . .;ϕ| |)=
R3
| |
i=1
|∇ϕi|2−
R3
V ρ+1
2D(ρ , ρ), (25) withρbeing defined as in (21). It is obvious that, for all ,
ERHEH, (26)
and thus
IRHIH. (27)
In the smeared nuclei case, the energy functional of the restricted Hartree model reads as follows
Em,RH(ϕ1;. . .;ϕ| |)=| |
i=1
R3
|∇ϕi|2−
R3
Vmρ+1
2D(ρ , ρ), (28) and the minimization problem can therefore be written in the following more concise form
Im,RH=inf | |
i=1
R3
|∇ϕi|2+1
2D(ρ−m , ρ−m )−1
2| |D(m, m); ϕi∈H1R3,
R3
ϕi2 =1,1i| |
. (29)
In view of the periodic problem that we have obtained in [11] for the TFW model, it is rather natural to introduce the following minimization problem, that we intend to relate with the Hartree model with fixed:
IperH =inf
EperH (ϕ);ϕ∈H1R3,
R3
|ϕ|2=1
, (30)
where the periodic energyEH is defined as follows EperH (ϕ)=
R3
|∇ϕ|2−1
2D|ϕ|2,|ϕ|2−
Q
Gρ+1
2DG(ρ , ρ), (31) with
ρ(x)=
k∈Z3
|ϕ|2(x+k), (32)
and the following notation that we shall adopt henceforth (in the spirit of the notation (13))
DG(f, f )=
Q
Q
f (x)G(x−y)f (y)dxdy. (33) We recall that Q denotes here and henceforth the unit cube ]−12,+12]3. On the other hand, for the restricted Hartree problem, we introduce the following minimization problem
IperRH=inf
EperRH(ρ); ρ0, √
ρ∈Hper1 (Q),
Q
ρ=1
, (34)
where we denote byHper1 (Q)the set of allQ-periodic functions inHloc1 (R3)and where the periodic energy functionalERHis given by
EperRH(ρ)=
Q
∇√ ρ2−
Q
Gρ+1
2DG(ρ , ρ). (35) It is easy to show that the minimization problem (34)–(35) admits a unique minimum (the same property will hold true in the smeared nuclei setting below). We now define the periodic H and RH problems in the smeared nuclei case.
Iperm,H =inf
Eperm,H(ϕ); ϕ∈H1R3,
R3
|ϕ|2=1
, (36)
where the periodic energyEperm,H is defined as follows Eperm,H(ϕ)=
R3
|∇ϕ|2−1
2D|ϕ|2,|ϕ|2+1
2DG(ρ−m , ρ−m)−1
2DG(m, m), (37) with the periodic densityρbeing related toϕthrough (32).
On the other hand, for the restricted Hartree problem, we introduce the following minimization problem
Iperm,RH=inf
Eperm,RH(ρ); ρ0, √
ρ∈Hper1 (Q),
Q
ρ=1
, (38)
Eperm,RH(ρ)=
Q
∇√ ρ2+1
2DG(ρ−m , ρ−m)−1
2DG(m, m). (39) The main purpose of Section 3 will be to prove the following result on the thermodynamic limit of the RH problem.
THEOREM 2.1 (Thermodynamic limit for the RH energy). – In the point nuclei case, we have
lim→∞
IRH
| | =IperRH+M 2 ,
where the constant Mis defined by (8). Likewise, in the smeared nuclei case, we have lim→∞
Im,RH
| | =Iperm,RH+M 2 , whereMis this time defined by
M=
Q×Q
m(x)m(y)G(x−y)−1/|x−y|dxdy. (40) We shall also make there some comments on this result.
As far as the Hartree model is concerned, we shall extensively present our point of view in Section 4, but let us already emphasize here that our main result will be the following one, which states that the minimization problem we have defined above is mathematically well-posed.
THEOREM 2.2 (Well-posedness of the H periodic problem). – The minimization problem defined by (30)–(31) (respectively by (36)–(37)) admits a minimum. In addition, any minimizing sequence of (30)–(31) (respectively (36)–(37)) is relatively compact in H1(R3), up to a translation.
Is is to be mentioned here that in the proof of the above theorem, we shall make use of the concentration-compactness method [34].
As announced in the introduction, we shall also see in Section 4 that, for a very particular choice of smeared nuclei, we are able to prove the convergence of the Hartree energy per unit volume to the periodic energy (30). We refer the reader to Proposition 4.1 below. We also prove in Section 4.4 the following.
PROPOSITION 2.1. – We assume that the Van Hove sequence satisfies lim→∞
| h|
| | Log| h| =0, ∀h >0, (41) where h is defined in Definition 1. We assume here that the unit cellQis a cube and that there exists a minimizer ϕper ∈H1(R3)ofIperH which shares the symmetries of the unit cube. Then,
lim sup
→∞
IH
| |IperH +M 2 , whereIperH is defined by (30)–(31).
As announced in the introduction, the sequel of this paper is devoted to the proofs of the above results. We shall also give some complements.
3. The restricted Hartree model
We devote this section to the thermodynamic limit problem of the so-called restricted Hartree model (RH model for short). We shall see that we shall be allowed to extend to this setting most of the methods introduced in [11] in order to prove that the TFW energy has a thermodynamic limit. Of course this study can be seen as a step towards the study of the complete Hartree model (H for short) that will be addressed in the following section. We shall see however that despite their relative formal resemblance, the RH model, on the one hand, and the Hartree model, on the other hand, do behave in a very different fashion, as far as the thermodynamic limit problem is concerned. For the time being, let us concentrate on the RH model.
Let us now recall the definition we have given in Section 2 above of the restricted Hartree model. For the sake of brevity, we shall only consider in this section the case of point nuclei. Actually, the case of smeared nuclei is easier to treat, and we leave it to the reader.
For every finite subset of Z3, the RH model is defined as follows:
IRH=inf
ERH(ϕ1;. . .;ϕ| |)+1 2U ;
∀1i| |, ϕi ∈H1R3,
R3
ϕi2 =1
, (42)
with
ERH(ϕ1;. . .;ϕ| |)=
R3
| |
i=1
|∇ϕi|2−
R3
V ρ+1 2
R3×R3
ρ(x)ρ(y)
|x−y| dxdy, (43)
ρ=|i=|1|ϕi|2, and where we recall that V (x) =y∈ 1
|x−y|. If we compare with the complete Hartree model given in (19)–(20), we may note that only the interaction between the electrons has been modified and has been replaced by a mean-field potential which is the same for each of the | |electrons. In other words, the self-interaction of each electron has been reincorporated into the energy functional.
We show now that, due to this modification, this infimum is the same as inf
ERH(ρ)+1
2U ; ρ0, √
ρ∈H1R3,
R3
ρ= | |
, (44)
with
ERH(ρ)=
R3
∇√ ρ2−
R3
V ρ+1 2
R3×R3
ρ(x)ρ(y)
|x−y| dxdy. (45) Indeed, we first recall from [35] that, on the one hand, the infimum defined by (44)–
(45) is achieved by a unique positive function ρRH (the uniqueness coming from the strict convexity of the functional ρ→ERH(ρ)defined by (45)). On the other hand, the
infimum in (42) is attained by| |positive functionsϕi, for 1i| |. In addition, for every 1i| |,ϕi satisfies
−ϕi−V ϕi+
ρ 1
|x| ϕi+θiϕi=0 on R3, (46) for someθi>0. This latter claim comes from the fact that, since ρ= | |, the positive part of the spherical average of the potential −V +(ρ |1x|), which is identically 0, lies in L3/2(R3). Then, we may apply a result of Lieb and Simon in [32]. Therefore, since ϕi >0 and since −V +(ρ |1x|) also belongs to Lpunif(R3), for some p > 32, ϕi is the (unique) positive normalized eigenfunction associated to the first eigenvalue of the operator −−V +(ρ |x1|) on R3, and the corresponding eigenspace is of dimension 1 (see, for example, [46]). We thus conclude that θ1 = · · · =θ| | and ϕ1= · · · =ϕ| |(=√1| |√
ρ). Then, returning to (46), we deduce thatϕ= √ρis a critical point forERH. Since the functionalρ→ERH(ρ)is strictly convex and sinceρsatisfies the right charge constraint, we conclude that ρ is the unique minimizer ofERH, that is ρRH. Our claim follows.
From now on, with a view to proving the existence of the thermodynamic limit for the energy per unit volume for the RH model, we shall essentially use the expression (45) for the energy and identifyIRH with (44). It is therefore to be emphasized that we deal with a sequence of minimization problems which are of density functional type: only the electronic density ρ appears in the minimization and not the electronic wavefunctions initially involved in (43). Consequently, we shall be able to use most of the machinery developed in [11] to treat the TFW model. As far as the thermodynamic limit for the energy per unit volume is concerned, this machinery (in particular the trick that consists of approximating the Coulomb problem by a problem where the interaction is of Yukawa type) will be effective and really allows us to determine the behaviour of ERH/| | (see Theorem 3.1 below). Unfortunately, we have not been able to use it in order to determine the behaviour of the density ρ , apart from some very basic results that will be mentioned below.
We shall relate the thermodynamic limit of the restricted Hartree model with the periodic minimization problem defined by
IperRH=inf
EperRH(ρ);ρ0, √
ρ∈Hper1 (Q),
Q
ρ =1
, (47)
where
EperRH(ρ)=
Q
∇√ ρ2−
Q
Gρ+1 2Q×Q
ρ(x)ρ(y)G(x−y)dxdy. (48) Before turning to the thermodynamic limit problem per se, let us first give some results on the existence and the uniqueness of the minimizer ofIperRH.
LEMMA 3.1 (Properties ofIperRH). – Let IperRH be defined by (47) and (48). Then, IperRH is achieved by a unique positive function ρper, uper= √ρper∈Hper1 (Q)∩L∞(R3), and
satisfies
−uper−Guper+
Q
G(x−y)ρper(y)dy uper+θperuper=0, on R3, (49) for some real numberθper.
Proof of Lemma 3.1. – The existence and the uniqueness of a minimizer ofIperRHfollows from the following observations. SinceG is periodic and since the functionG−|1x| is continuous and bounded on Q, it is easy to check that IperRH is finite. Indeed, on the one hand, it is easily seen , by using for example the Fourier series expansion of the periodic potentialG(see [32]), that the quadratic formf →DG(f, f )is non-negative.
On the other hand, sinceGis inL3/2unif(R3), for every ε >0, there is a positive constant k(ε) such that we may decompose G into G=G1+G2 with G1L∞(Q)k(ε) and G2L3/2(Q)ε. Now let ρ 0 be such that u≡ √ρ∈Hper1 (Q). We first notice that 0<Qu1 because Qu2=1, and from Schwarz’s inequality. Therefore, we have, using first Hölder’s, and then Sobolev–Poincaré’s inequalities,
EperH (ρ)
Q
|∇u|2−
Q
G(x)u2(x)dx
Q
|∇u|2−k(ε)
Q
u2−εu2L6(Q)
(1−2ε)
Q
|∇u|2−k(ε)−Cε,
for some positive constantC, that is independent ofεandu. We conclude by choosingε small enough.
By the way, the same argument shows that every minimizing sequence ρn ofIperH is such thatun= √ρnis bounded inHper1 (Q). Then, extracting a subsequence if necessary, we may assume that un converges weakly in Hper1 (Q), strongly in Lpunif(R3) for all 1p <6 (from Rellich’s Theorem) and almost everywhere on R3. The limit is then a minimizer ofIperH. The uniqueness of the minimizer follows from the strict convexity of the functional.
In addition, since G is in Lqunif(R3) for all 1q < 3, it is clear from (49) that
−uper is inLpunif for every 1p <2. Thus,u∈Wunif2,p. In particular, from Sobolev’s embeddings,u∈L∞(R3). In fact, by a standard bootstrap argument,uis inWunif2,p∩C0, α, for every 1p <3 and 0< α <1. ✷
Let us turn now to the thermodynamic limit problem we are interested in and prove first the following:
LEMMA 3.2. – For every Van Hove sequence( ), we have lim sup
→∞
IRH
| | IperRH+M
2 . (50)
Proof of Lemma 3.2. – The proof is immediate once we have noticed that, for allε >0,
IRHI , εTFW, (51)
where the notation I , εTFW stands for the usual TFW problem we have studied in [11], withεas a coefficient in front of the Thomas–Fermi termR3ρ5/3in the definition of the TFW functional; namely
ETFW ,ε (ρ)=
R3
∇√ ρ2+ε
R3
ρ5/3−
R3 k∈
1
|x−k| ρ(x)dx +1
2
R3×R3
ρ(x)ρ(y)
|x−y| dxdy, (52)
I ,εTFW=inf
ETFW ,ε (ρ)+1 2
y=z∈
1
|y−z|;ρ0, √
ρ∈H1R3,
R3
ρ= | |
. (53) Next, in view of the results of [11], we obtain from (51), and for everyε >0,
lim sup
→∞
IH
| | lim
→∞
I , εTFW
| | =Iper,εTFW+M
2 , (54)
where, obviously,Iper,εTFW is the periodic TFW model with a multiplicative parameterεin front of the termQρ5/3in the definition of the TFW periodic functional; namely
Iper,εTFW=inf
ETFWper,ε(ρ); ρ0, √
ρ∈Hper1 (Q),
Q
ρ=1
, (55)
Eper,εTFW(ρ)=
Q
∇√ ρ2+ε
Q
ρ5/3−
Q
ρ(x)G(x)dx +1
2Q×Q
ρ(x)ρ(y)G(x−y)dxdy. (56) Assertion (50) follows now by letting ε go to 0 in (54), and by comparing with the definition (47) ofIperRH. ✷
We next prove the existence of a bound from below for the energy per unit volume in the RH case.
LEMMA 3.3. – For every Van Hove sequence( ), we have lim inf
→∞
IRH
| | IperRH+M
2 . (57)
Proof of Lemma 3.3. – Our strategy of proof will consist of comparing from below IRH with the corresponding minimization problem where the Coulomb potential has been replaced by a Yukawa potential exp(|−xa||x|), a >0, and then letting a go to 0. Let us recall that the same strategy has already been used in [11] in the TFW setting. We shall therefore only sketch the main lines of the proof and refer the reader to [11] for the details.
We thus define, for everya >0, Ia=inf
Ea(ρ)+1
2Ua; ρ0, √
ρ∈H1R3,
R3
ρ= | |
, (58)
with
Ea(ρ)=
R3
∇√ ρ2−
R3
Vaρ+1 2
R3×R3
ρ(x)ρ(y)Va(x−y)dxdy, (59)
Va(x)=exp(−a|x|)
|x| , Va(x)=
y∈
Va(x−y), and Ua =
y,z∈ y=z
Va(y−z).
It is clear that we may choosea small enough such that Ia is achieved for all finite subset of Z3. In addition, by using the methods of Chapter 2 of [11] for the upper limit and the ones of Chapter 3 of [11] for the lower limit, it is easy to check that
lim→∞
Ia
| |=Ipera (µa), (60) for any Van Hove sequence( ), whereµaandIpera (µa)are defined just below. We set
Ipera (µa)=inf
Epera (ρ)+1
2U∞a;ρ0, √
ρ∈Hper1 (Q),
Q
ρ=µa
,
with
Epera (ρ)=
Q
∇√ ρ2−
Q
V∞a(x)ρ(x)dx+1 2Q×Q
ρ(x)ρ(y)V∞a(x−y)dxdy,
V∞a(x)=
y∈Z3
Va(x−y), and U∞a =
y,z∈Z3 y=z
Va(y−z).
Finally the massµais defined as follows. We denote byρpera the unique minimizer of Epera on the set{ρ0,√ρ∈Hper1 (Q)}. Then, we defineµa=min(1,Qρpera ). (All these definitions are justified in [11].) Arguing as in Chapter 2 of [11], we may prove that
alim→0+µa=1,
and that
lim
a→0+Ipera (µa)=IperRH+M
2 . (61)
To conclude, we argue now as in Chapter 3 of [11], to check that IRH
| | Ia
| |−C a,
for some positive constant C that is independent of . Next, we let go to infinity in the above inequality and use (60) to obtain
lim inf
→∞
IRH
| | Ipera (µa)−Ca.
(57) then follows by lettingago to 0 and by using (61). ✷
Remark 3.1. – In the case when the unit cell is a cube, it is possible to prove the above lemma by a different argument which does not use the comparison with a Yukawa potential. Indeed, as in [11], we may use the ∼-transform trick and prove directly the lower bound. Of course, this argument relies upon the convexity of the RH functional with respect to the electronic density.
As a consequence of (50), we may prove that
COROLLARY 3.1 (Compactness). – Letρ be the minimizer ofIRH, then 1
| |
( )c
ρ →0, as → ∞.
The analogous result holds true in the Hartree setting, and a proof is sketched in this setting (see the proof of Lemma 4.3 below).
Remark 3.2. – This property means that, asymptotically, the | | electrons remain in ( ); that is, in a box of volume | |. In other words, we could also say that no electrons have escaped to “infinity”; this is the reason why this property is referred to as
“compactness” in [11].
Finally, collecting Lemma 3.2 and Lemma 3.3, we have proved the following
THEOREM 3.1 (Thermodynamic limit for the energy in the RH model). – For every Van Hove sequence ( ),
lim→∞
IRH
| | =IperRH+M
2 . (62)
Let us make some comments. Having proved the existence of the thermodynamic limit for the energy per unit volume for the RH model, we may prove as in Chapter 5 of [11], some preliminary convergence results concerning the convergence of the densities. In particular, we may show that the∼-transform ofρ ,ρ˜ , converges to the minimizerρper
